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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font> being   <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">W2</font>, <font color="Olive" title="b3">W3</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> holds <br/><font color="Olive" title="b1">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b3">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b2">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b3">W3</font></span><br/><b>let </b><font color="Maroon" title="c2">W1</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W2</font>, <font color="Olive" title="b2">W3</font> being <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b1">W2</font> holds <br/><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b2">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Olive" title="b1">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Olive" title="b2">W3</font></span><br/><b>let </b><font color="Maroon" title="c3">W2</font>, <font color="Maroon" title="c4">W3</font> be  <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>  <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> implies <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font> )</span><br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W3</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self" title="RUSUB_2:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W3</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self" title="RUSUB_2:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E8"><i><font color="Green" title="E8">Lm3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W3</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self" title="RUSUB_2:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font></span>)</span>
<b>by </b><i><a class="txt" href="rusub_2.html#E6"><i><font color="Green" title="E6">Lm1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c3">W2</font></span>)</span> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font>
<b>by </b><i><a class="ref" href="rusub_2.html#T6" target="_self" title="RUSUB_2:th.6">Th6</a></i>
<br/>.= 
<font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font>
<b>by </b><i><a class="txt" href="rusub_2.html#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="rusub_2.html#T8" target="_self" title="RUSUB_2:th.8">Th8</a></i>
;<br/>
<b>hence </b><a NAME="E3:33"/>
<font color="Maroon" title="c2">W1</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font> <a href="rusub_2.html#K1" title="RUSUB_2:func.1">+</a> <font color="Maroon" title="c4">W3</font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T8" target="_self" title="RUSUB_2:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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